Local rigidity of Lyapunov spectrum for toral automorphisms

We study the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L⁴ is irreducible over ℚ. We consider a volume-preserving C¹-small perturbation f of L. We show that if Lyapunov exponents of f with respect to the volume are the same as Lyapunov exponents of L, then f is C^1+Hölder conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle.